ECE 421  Sum 2010
Notes Set 8:
Signal Quantization
1
INTRODUCTION
In digital dignal processing we frequently use digital algorithms which compute values
for discrete points in time, or for discrete points in “space”, like in digital imaging. The
input to these algorithms is often the sampled data from AnalogtoDigital Converters.
However, when we implement these signal processing algorithms on a DSPchip (which is
a computer) or in digital hardware, then there are other effects become very important as
well. Two of these effects are signal quantization and finiteprecision arithmetic effects.
SIGNAL QUANTIZATION:
Consider the case in which we want to implement an algorithm (like a filter) on a DSP
chip. Like a computer, the DSP chip has a finite number of bits per word, both in
instructions and data representation. The AnalogtoDigital Converter (ADC) trans
forms the continuum of values possessed by the analog signal into a finite number
of possible values. Each analog sample is now represented by a finite number of
bits, like 16 bits or 32 bits, and this introduces noise. We will call this case signal
quantization noise and it is the topic of this set of Notes.
FINITEPRECISION EFFECTS:
The DSP chip must implement the algorithm using computer structures such as ac
cumulator, storage, bus transfers, etc. Consider the case of multiplying two 18bit
words and storing the result in a 18bit memory. This multiplication requires 32bits
for full accuracy, so we would need a 32bit memory. If we must store the multipli
cation result in a 18bit memory this means we have to reduce the precision of the
product back to 18bits, and this also introduces noise. This phenomena is studied in
the next set of Notes
These are the two quantization processes we will study in this course. We next need to
quantify the concept of a digital word having a finite number of bits.
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ECE 421  Sum 2010
Notes Set 8:
Signal Quantization
2
DIGITAL WORDS
Many DSP chips may use 18bit words or 32bit words, or even longer words. However, for
simplicity we will often use 3 or 4 bit words in our examples. This will make the concepts
easier to understand but the principles will extend to any wordlength.
SIGNMAGNITUDE FORMAT :
Many digital devices use a 2’scomplement number representation for actual computations.
However, the signmagnitude is perhaps easier for us to understand when we are learning
concepts. We will refer to algorithms implemented using “
b
bit arithmetic”, where
b
is
the number of bits excluding the signbit. As an example of using this convention, the
representation of two
b
= 4bit data words is shown below:
SignMagnitude
Decimal
Integer
0
1
0
0
0
0 5000
8
0
0
1
1
1
0 4375
7
1
0
1
1
1
0 4375
7
The
in the above is the “binary point”. It is not a part of the data representation, but
is shown for our benefit. The sign bit is to the left of the binary point: a 0 in the sign bit
implies positive and 1 implies negative. Note that the digital words above actually require 5
bits to store in memory, but we will refer to this as 4bit (arithmetic) words. The magnitude
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 Summer '08
 HALLEN
 Digital Signal Processing, Algorithms, Signal Processing, Probability theory, Signal Quantization, signmagnitude roundoff quantizer

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